Generalization and variations of Pellet's theorem for matrix polynomials

TL;DR

This paper extends Pellet's theorem to matrix polynomials using a generalized Rouché theorem, providing bounds on eigenvalues and variations to address limitations of the original theorem.

Contribution

It introduces a generalized matrix Pellet's theorem and its variations, enhancing eigenvalue bounding techniques for matrix polynomials.

Findings

Derived a matrix version of Pellet's theorem based on a generalized Rouché theorem.

Proposed variations of the theorem to handle cases where the original cannot be applied.

Provided methods to generate upper, lower, and internal eigenvalue bounds.

Abstract

We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of the theorem are suggested to try and overcome situations where Pellet's theorem cannot be applied.